I understand, and often use, the shortcut Comman-Shift-S to sync with Dropbox, but am unsure what Command-S (Save) does. The app has crashed three times, without me doing anything to “save” it, and each time all my text was there when I launched the app again.

Should I be using it or does Scrivener “save” the document (I assume locally, on my iPad) frequently itself?

Writing on a late-2013 MacBook Pro running High Sierra and a 2017 12.9" iPad Pro.

I'd say you don't have to use it. All of us that don't use an external keyboard never save anything because, as far as I know, there are no keyboard shortcuts on the internal keyboard on an iPad or iPhone. Scrivener keeps it all in the memory of the iOS device, and sync to Dropbox when we tell it to by tapping on the sync icon.

I imagine that it's built into Apple's TextKit, and Keith decided that removing (or more likely, hiding) it would take unneeded work. But since I'm not on the L&L team but a mere user like yourself, I only speculate based on years misspent as a software developer.

On the Mac, Command-S can be used to trigger either a snapshot of the current document or a backup of the project, via the snapshot/backup on manual save options. (Found in the Scrivener -> Preferences panel.) That option isn't available for iOS, but many users have the Cmd-S habit ingrained and would miss it if it were gone or did something else.

On both platforms, manually saving is generally unnecessary because of Scrivener's autosave features.